In this paper, we are concerned in obtaining multicast trees in packet-switched networks such as ATM nets, when there exist constraints on the packet (cell)-replication capabilities of the individual switching nodes. This problem can be formulated as the Steiner tree problem with degree bounds on the nodes, so we call it the Degree-Constrained Steiner Tree problem (DCST). Four heuristic algorithms are proposed: the first is a combined version of two well-known Steiner tree algorithms, heuristic Naive and the shortest path heuristic (SPH), and the second is a relaxation algorithm based on a mathematical formulation of the DCST, and the last two use a tree reconfiguration scheme based on the concept of 'logical link. ' We experimentally compare our algorithms with the previous ones in three respects; number of solved instances, objective value or tree cost, and computation time. The experimental results show that there are few instances unsolved by our algorithms, and the objective values are mostly within 5% of optimal. Computation times are also acceptable.

This paper describes the waveform relaxation (WR) algorithm with the under relaxation method based on the virtual state formulation (VSF) technique and the effect of multirate behavior in this algorithm. First, we present the virtual state relaxation method using VSF technique. Next, we introduce the VSF method into WR algorithm in order to exploit the multirate behavior. Furthermore, we construct the relaxation-based circuit simulator DESIRE2 and apply this simulator to the transient analysis of MOS circuits. Finally, we show that the present technique enables to use efficiently the multirate integration method in VSR and reduce the total simulation time without losing the waveform accuracy.